A Terse Introduction To Lebesgue Integration
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A Terse Introduction To Lebesgue Integration
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Author : John M. Franks
language : en
Publisher: American Mathematical Soc.
Release Date : 2009
A Terse Introduction To Lebesgue Integration written by John M. Franks and has been published by American Mathematical Soc. this book supported file pdf, txt, epub, kindle and other format this book has been release on 2009 with Mathematics categories.
This book provides a student's first encounter with the concepts of measure theory and functional analysis. Its structure and content reflect the belief that difficult concepts should be introduced in their simplest and most concrete forms. Despite the use of the word 'terse' in the title, this text might also have been called ""A (Gentle) Introduction to Lebesgue Integration"". It is terse in the sense that it treats only a subset of those concepts typically found in a substantial graduate-level analysis course. The book emphasizes the motivation of these concepts and attempts to treat them simply and concretely. In particular, little mention is made of general measures other than Lebesgue until the final chapter and attention is limited to R as opposed to Rn. After establishing the primary ideas and results, the text moves on to some applications. Chapter 6 discusses classical real and complex Fourier series for L2 functions on the interval and shows that the Fourier series of an L2 function converges in L2 to that function. Chapter 7 introduces some concepts from measurable dynamics. The Birkhoff ergodic theorem is stated without proof and results on Fourier series from chapter 6 are used to prove that an irrational rotation of the circle is ergodic and that the squaring map on the complex numbers of modulus 1 is ergodic. This book is suitable for an advanced undergraduate course or for the start of a graduate course. The text presupposes that the student has had a standard undergraduate course in real analysis.
The Stieltjes Integral
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Author : Gregory Convertito
language : en
Publisher: CRC Press
Release Date : 2023-02-28
The Stieltjes Integral written by Gregory Convertito and has been published by CRC Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 2023-02-28 with Mathematics categories.
The Stieltjes Integral provides a detailed, rigorous treatment of the Stieltjes integral. This integral is a generalization of the Riemann and Darboux integrals of calculus and undergraduate analysis, and can serve as a bridge between classical and modern analysis. It has applications in many areas, including number theory, statistics, physics, and finance. It begins with the Darboux integral, builds the theory of functions of bounded variation, and then develops the Stieltjes integral. It culminates with a proof of the Riesz representation theorem as an application of the Stieltjes integral. For much of the 20th century the Stjeltjes integral was a standard part of the undergraduate or beginning graduate student sequence in analysis. However, the typical mathematics curriculum has changed at many institutions, and the Stieltjes integral has become less common in undergraduate textbooks and analysis courses. This book seeks to address this by offering an accessible treatment of the subject to students who have had a one semester course in analysis. This book is suitable for a second semester course in analysis, and also for independent study or as the foundation for a senior thesis or Masters project. Features: Written to be rigorous without sacrificing readability. Accessible to undergraduate students who have taken a one-semester course on real analysis. Contains a large number of exercises from routine to challenging.
Introduction To Representation Theory
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Author : Pavel I. Etingof
language : en
Publisher: American Mathematical Soc.
Release Date : 2011
Introduction To Representation Theory written by Pavel I. Etingof and has been published by American Mathematical Soc. this book supported file pdf, txt, epub, kindle and other format this book has been release on 2011 with Mathematics categories.
Very roughly speaking, representation theory studies symmetry in linear spaces. It is a beautiful mathematical subject which has many applications, ranging from number theory and combinatorics to geometry, probability theory, quantum mechanics, and quantum field theory. The goal of this book is to give a ``holistic'' introduction to representation theory, presenting it as a unified subject which studies representations of associative algebras and treating the representation theories of groups, Lie algebras, and quivers as special cases. Using this approach, the book covers a number of standard topics in the representation theories of these structures. Theoretical material in the book is supplemented by many problems and exercises which touch upon a lot of additional topics; the more difficult exercises are provided with hints. The book is designed as a textbook for advanced undergraduate and beginning graduate students. It should be accessible to students with a strong background in linear algebra and a basic knowledge of abstract algebra.
A Primer On The Calculus Of Variations And Optimal Control Theory
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Author : Mike Mesterton-Gibbons
language : en
Publisher: American Mathematical Soc.
Release Date : 2009
A Primer On The Calculus Of Variations And Optimal Control Theory written by Mike Mesterton-Gibbons and has been published by American Mathematical Soc. this book supported file pdf, txt, epub, kindle and other format this book has been release on 2009 with Mathematics categories.
The calculus of variations is used to find functions that optimize quantities expressed in terms of integrals. Optimal control theory seeks to find functions that minimize cost integrals for systems described by differential equations. This book is an introduction to both the classical theory of the calculus of variations and the more modern developments of optimal control theory from the perspective of an applied mathematician. It focuses on understanding concepts and how to apply them. The range of potential applications is broad: the calculus of variations and optimal control theory have been widely used in numerous ways in biology, criminology, economics, engineering, finance, management science, and physics. Applications described in this book include cancer chemotherapy, navigational control, and renewable resource harvesting. The prerequisites for the book are modest: the standard calculus sequence, a first course on ordinary differential equations, and some facility with the use of mathematical software. It is suitable for an undergraduate or beginning graduate course, or for self study. It provides excellent preparation for more advanced books and courses on the calculus of variations and optimal control theory.
Lectures On Fractal Geometry And Dynamical Systems
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Author : Ya. B. Pesin
language : en
Publisher: American Mathematical Soc.
Release Date : 2009
Lectures On Fractal Geometry And Dynamical Systems written by Ya. B. Pesin and has been published by American Mathematical Soc. this book supported file pdf, txt, epub, kindle and other format this book has been release on 2009 with Mathematics categories.
Both fractal geometry and dynamical systems have a long history of development and have provided fertile ground for many great mathematicians and much deep and important mathematics. These two areas interact with each other and with the theory of chaos in a fundamental way: many dynamical systems (even some very simple ones) produce fractal sets, which are in turn a source of irregular 'chaotic' motions in the system. This book is an introduction to these two fields, with an emphasis on the relationship between them. The first half of the book introduces some of the key ideas in fractal geometry and dimension theory - Cantor sets, Hausdorff dimension, box dimension - using dynamical notions whenever possible, particularly one-dimensional Markov maps and symbolic dynamics. Various techniques for computing Hausdorff dimension are shown, leading to a discussion of Bernoulli and Markov measures and of the relationship between dimension, entropy, and Lyapunov exponents. In the second half of the book some examples of dynamical systems are considered and various phenomena of chaotic behaviour are discussed, including bifurcations, hyperbolicity, attractors, horseshoes, and intermittent and persistent chaos. These phenomena are naturally revealed in the course of our study of two real models from science - the FitzHugh - Nagumo model and the Lorenz system of differential equations. This book is accessible to undergraduate students and requires only standard knowledge in calculus, linear algebra, and differential equations. Elements of point set topology and measure theory are introduced as needed. This book is a result of the MASS course in analysis at Penn State University in the fall semester of 2008.
Geometries
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Author : Alekseĭ Bronislavovich Sosinskiĭ
language : en
Publisher: American Mathematical Soc.
Release Date : 2012
Geometries written by Alekseĭ Bronislavovich Sosinskiĭ and has been published by American Mathematical Soc. this book supported file pdf, txt, epub, kindle and other format this book has been release on 2012 with Mathematics categories.
The book is an innovative modern exposition of geometry, or rather, of geometries; it is the first textbook in which Felix Klein's Erlangen Program (the action of transformation groups) is systematically used as the basis for defining various geometries. The course of study presented is dedicated to the proposition that all geometries are created equal--although some, of course, remain more equal than others. The author concentrates on several of the more distinguished and beautiful ones, which include what he terms ``toy geometries'', the geometries of Platonic bodies, discrete geometries, and classical continuous geometries. The text is based on first-year semester course lectures delivered at the Independent University of Moscow in 2003 and 2006. It is by no means a formal algebraic or analytic treatment of geometric topics, but rather, a highly visual exposition containing upwards of 200 illustrations. The reader is expected to possess a familiarity with elementary Euclidean geometry, albeit those lacking this knowledge may refer to a compendium in Chapter 0. Per the author's predilection, the book contains very little regarding the axiomatic approach to geometry (save for a single chapter on the history of non-Euclidean geometry), but two Appendices provide a detailed treatment of Euclid's and Hilbert's axiomatics. Perhaps the most important aspect of this course is the problems, which appear at the end of each chapter and are supplemented with answers at the conclusion of the text. By analyzing and solving these problems, the reader will become capable of thinking and working geometrically, much more so than by simply learning the theory. Ultimately, the author makes the distinction between concrete mathematical objects called ``geometries'' and the singular ``geometry'', which he understands as a way of thinking about mathematics. Although the book does not address branches of mathematics and mathematical physics such as Riemannian and Kahler manifolds or, say, differentiable manifolds and conformal field theories, the ideology of category language and transformation groups on which the book is based prepares the reader for the study of, and eventually, research in these important and rapidly developing areas of contemporary mathematics.
The Erdos Distance Problem
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Author : Julia Garibaldi
language : en
Publisher: American Mathematical Soc.
Release Date : 2011
The Erdos Distance Problem written by Julia Garibaldi and has been published by American Mathematical Soc. this book supported file pdf, txt, epub, kindle and other format this book has been release on 2011 with Mathematics categories.
Introduces the reader to the techniques, ideas, and consequences related to the Erdős problem. The authors introduce these concepts in a concrete and elementary way that allows a wide audience to absorb the content and appreciate its far-reaching implications. In the process, the reader is familiarized with a wide range of techniques from several areas of mathematics and can appreciate the power of the resulting symbiosis.
Volterra Adventures
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Author : Joel H. Shapiro
language : en
Publisher: American Mathematical Soc.
Release Date : 2018-06-14
Volterra Adventures written by Joel H. Shapiro and has been published by American Mathematical Soc. this book supported file pdf, txt, epub, kindle and other format this book has been release on 2018-06-14 with Mathematics categories.
This book introduces functional analysis to undergraduate mathematics students who possess a basic background in analysis and linear algebra. By studying how the Volterra operator acts on vector spaces of continuous functions, its readers will sharpen their skills, reinterpret what they already know, and learn fundamental Banach-space techniques—all in the pursuit of two celebrated results: the Titchmarsh Convolution Theorem and the Volterra Invariant Subspace Theorem. Exercises throughout the text enhance the material and facilitate interactive study.
Random Walk And The Heat Equation
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Author : Gregory F. Lawler
language : en
Publisher: American Mathematical Soc.
Release Date : 2010-11-22
Random Walk And The Heat Equation written by Gregory F. Lawler and has been published by American Mathematical Soc. this book supported file pdf, txt, epub, kindle and other format this book has been release on 2010-11-22 with Mathematics categories.
The heat equation can be derived by averaging over a very large number of particles. Traditionally, the resulting PDE is studied as a deterministic equation, an approach that has brought many significant results and a deep understanding of the equation and its solutions. By studying the heat equation and considering the individual random particles, however, one gains further intuition into the problem. While this is now standard for many researchers, this approach is generally not presented at the undergraduate level. In this book, Lawler introduces the heat equations and the closely related notion of harmonic functions from a probabilistic perspective. The theme of the first two chapters of the book is the relationship between random walks and the heat equation. This first chapter discusses the discrete case, random walk and the heat equation on the integer lattice; and the second chapter discusses the continuous case, Brownian motion and the usual heat equation. Relationships are shown between the two. For example, solving the heat equation in the discrete setting becomes a problem of diagonalization of symmetric matrices, which becomes a problem in Fourier series in the continuous case. Random walk and Brownian motion are introduced and developed from first principles. The latter two chapters discuss different topics: martingales and fractal dimension, with the chapters tied together by one example, a random Cantor set. The idea of this book is to merge probabilistic and deterministic approaches to heat flow. It is also intended as a bridge from undergraduate analysis to graduate and research perspectives. The book is suitable for advanced undergraduates, particularly those considering graduate work in mathematics or related areas.
Computability Theory
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Author : Rebecca Weber
language : en
Publisher: American Mathematical Soc.
Release Date : 2012
Computability Theory written by Rebecca Weber and has been published by American Mathematical Soc. this book supported file pdf, txt, epub, kindle and other format this book has been release on 2012 with Mathematics categories.
What can we compute--even with unlimited resources? Is everything within reach? Or are computations necessarily drastically limited, not just in practice, but theoretically? These questions are at the heart of computability theory. The goal of this book is to give the reader a firm grounding in the fundamentals of computability theory and an overview of currently active areas of research, such as reverse mathematics and algorithmic randomness. Turing machines and partial recursive functions are explored in detail, and vital tools and concepts including coding, uniformity, and diagonalization are described explicitly. From there the material continues with universal machines, the halting problem, parametrization and the recursion theorem, and thence to computability for sets, enumerability, and Turing reduction and degrees. A few more advanced topics round out the book before the chapter on areas of research. The text is designed to be self-contained, with an entire chapter of preliminary material including relations, recursion, induction, and logical and set notation and operators. That background, along with ample explanation, examples, exercises, and suggestions for further reading, make this book ideal for independent study or courses with few prerequisites.